Certain infinite Markov chains and sequential decoding

Abstract

It is shown that the sequential decoding of rate one-half convolutional codes leads to a special type of infinite Markov chain in which only one transition of one step toward the origin state is permitted but any number of transitions away from the origin state are permitted. It is shown that such chains have the singular property that the stationary state probabilities π 0,π 1,π 2,… can be calculated successively without summing any infinite series or solving for any eigenvalues. It is further shown that the analogous infinite Markov chains in which a single transition of one step away from the origin is permitted are also related to a problem in sequential decoding and also have a singular property, namely that π i = (1 − β) β i where β is the unique real eigenvalue of the characteristic polynomial such that 0 < β < 1.